let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cos * exp_R ) implies ( cos * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * exp_R ) `| Z) . x = - ((exp_R . x) * (sin . (exp_R . x))) ) ) )
A1: for x being Real st x in Z holds
cos * exp_R is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cos * exp_R is_differentiable_in x )
assume x in Z ; :: thesis: cos * exp_R is_differentiable_in x
( exp_R is_differentiable_in x & cos is_differentiable_in exp_R . x ) by SIN_COS:68, SIN_COS:70;
hence cos * exp_R is_differentiable_in x by FDIFF_2:13; :: thesis: verum

end;

assume A2: Z c= dom (cos * exp_R ) ; :: thesis: ( cos * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * exp_R ) `| Z) . x = - ((exp_R . x) * (sin . (exp_R . x))) ) )
then A3: cos * exp_R is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((cos * exp_R ) `| Z) . x = - ((exp_R . x) * (sin . (exp_R . x)))

proof

let x be Real; :: thesis: ( x in Z implies ((cos * exp_R ) `| Z) . x = - ((exp_R . x) * (sin . (exp_R . x))) )
( exp_R is_differentiable_in x & cos is_differentiable_in exp_R . x ) by SIN_COS:68, SIN_COS:70;
then A4: diff (cos * exp_R ),x = (diff cos ,(exp_R . x)) * (diff exp_R ,x) by FDIFF_2:13
.= (- (sin . (exp_R . x))) * (diff exp_R ,x) by SIN_COS:68
.= (- (sin . (exp_R . x))) * (exp_R . x) by SIN_COS:70
.= - ((exp_R . x) * (sin . (exp_R . x))) ;
assume x in Z ; :: thesis: ((cos * exp_R ) `| Z) . x = - ((exp_R . x) * (sin . (exp_R . x)))
hence ((cos * exp_R ) `| Z) . x = - ((exp_R . x) * (sin . (exp_R . x))) by A3, A4, FDIFF_1:def 8; :: thesis: verum

end;

hence ( cos * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * exp_R ) `| Z) . x = - ((exp_R . x) * (sin . (exp_R . x))) ) ) by A2, A1, FDIFF_1:16; :: thesis: verum